Saturday 6th April, Northern and Transpennine strikes have been announced
I need to travel from Lancaster to Manchester Piccadilly, however Northern and Transpennine both run these services.
Is it possible to use my ticket on Avanti West Coast services to Crewe and then from Crewe to Manchester Piccadilly, as this is the best route available on that day now?
If you hadn't already bought a ticket before the strikes were announced, then unfortunately you can't force the railway to provide a service that it effectively doesn't want to - and so you'd have to buy split tickets that allow you to travel via Crewe or another route that has services.
If, on the other hand, you already held tickets when the strike was announced, the railway remains obliged to get you to your destination. If there are no train services or replacement buses on the usual permitted routes (as will almost certainly be the case as it's an ASLEF driver strike), they are obliged to allow you to travel via alternative routes to complete your journey.
As you would presumably otherwise be "stranded", condition 28.2 of the National Rail Conditions of Travel is engaged. This means that the above obligation applies not only in relation to the train company/ies you were intending to travel with, but in fact any train company that is in a position to assist you.
Avanti would fall into this category as they are still running services and you can use them to make the journey, e.g. via Crewe, as you've said.
Unfortunately, in practical terms it is possible that Avanti staff will wrongly claim that they aren't obliged to assist you. They may try and charge you for a new ticket. If this happens, the best course of action is to pay what they're demanding and then contact Avanti afterwards to claim this back.